Proposition 16

— Let A be a convex subset, with at least one interior point $x_0$, of a topological vector space E. For any point $x \in \overline{A}$, every point of the open segment with end points $x_0, x$ lies in the interior of A.

For any point y of this segment, let f be the homothety of centre y and ratio $\lambda < 0$, which transforms $x_0$ into x. If V is an open neighbourhood of $x_0$ contained in A, then $f(V)$ is a neighbourhood of x and therefore contains a point $f(z) \in A$; now

hence $y - f(z) = \frac{\lambda}{\lambda - 1} (z - f(z))$, so that y is transformed into z by the homothety g, of centre $f(z)$ and ratio $\mu = \lambda / (\lambda - 1)$; since $0 < \mu < 1$, g transforms V into a neighbourhood of 0 contained in A. The proposition is proved.